[Proof]
Motivated by Lemma 8 of [1]}, for \( \Phi (s) = s -1 - \ln s \) , \( s>0 \) , noticing that
\(\left[ R \Theta \Phi \left( \frac{v}{V} \right) \right]_t= -R \Theta \left( \frac{1}{v} - \frac{1}{V} \right) \phi _t - \frac{P \phi ^2}{v V}V_t + R \Theta _t \Phi \left( \frac{v}{V} \right)\) 
